Ask Ethan: How Do Quantum Fields Create Particles?

At the high temperatures achieved in the very young Universe, not only can particles and photons be spontaneously created, given enough energy, but also antiparticles and unstable particles as well, resulting in a primordial particle-and-antiparticle soup. Yet even with these conditions, only a few specific states, or particles, can emerge.

Brookhaven National Laboratory

What is our Universe made out of? At a fundamental level, to the best of our knowledge, the answer is simple: particles and fields. The type of matter that makes up humans, Earth, and all the stars, for example, is all composed of the known particles of the Standard Model. Dark matter is theorized to be a particle, while dark energy is theorized to be a field inherent to space itself. But all the particles that exist, at the core of their nature, are just excited quantum fields themselves. What gives them the properties that they have? That's the topic of this week's question, coming to us from Richard Hunt, who wants to know:

I have a question about Quantum fields. If we model particle properties as excitations of various independent fields (Higgs field for mass, EM field for charge etc) then what causes these excitation waves to travel around together? Is there really some kind of particle entity underlying these waves?

In other words: what makes a particle have the properties that it does? Let's take a deep look.

The particles and antiparticles of the Standard Model have now all been directly detected, with the last holdout, the Higgs Boson, falling at the LHC earlier this decade. All of these particles can be created at LHC energies, and the masses of the particles lead to fundamental constants that are absolutely necessary to describe them fully. These particles can be well-described by the physics of the quantum field theories underlying the Standard Model, but whether they are fundamental is not yet known.

E. Siegel / Beyond The Galaxy

The particles that we know of have traits that appear to be inherent to them. All particles of the same type — electrons, muons, up quarks, Z-bosons, etc. — are, at some level, indistinguishable from one another. They all have a slew of properties that all other particles of the same type share, including:

mass,

electric charge,

weak hypercharge,

spin (inherent angular momentum),

color charge,

baryon number,

lepton number,

lepton family number,

and more. Some particles have a value of zero for many of these quantities; others have non-zero values for almost all of them. But somehow, every particle that exists contains all of these particular, intrinsic properties bound together in a single, stable, "quantum state" we call a particular particle.

The rest masses of the fundamental particles in the Universe determine when and under what conditions they can be created. The more massive a particle is, the less time it can spontaneously be created for in the early Universe. The properties of particles, fields, and spacetime are all required to describe the Universe we inhabit.

Fig. 15-04a from universe-review.ca

Underlying all of it, there are a variety of fields that exist in the Universe. There's the Higgs field, for example, which is a quantum field that permeates all of space. The Higgs is a relatively simple example of a field, even though the particle that arose from its behavior — the Higgs boson — was the last one ever to be discovered. The electromagnetic (QED) field and color-charge (QCD) field, among others, are also fundamental quantum fields.

Here's how it works: the field exists everywhere in space, even when there are no particles present. The field is quantum in nature, which means it has a lowest-energy state that we call the zero-point energy, whose value may or may not be zero. Across different locations in space and time, the value of the field fluctuates, just like all quantum fields do. The quantum Universe, to the best of our understanding, has rules governing its fundamental indeterminism.

Visualization of a quantum field theory calculation showing virtual particles in the quantum vacuum. Even in empty space, this vacuum energy is non-zero, but without specific boundary conditions, individual particle properties will not be constrained.

Derek Leinweber

So if everything is fields, then what is a particle? You may have heard a phrase before: that particles are excitations of quantum fields. In other words, these are quantum fields not in their lowest-energy — or zero-point — state, but in some higher-energy state. But exactly how this works is a bit tricky.

Up until this point, we've been thinking of fields in terms of empty space: the quantum fields we're discussing exist everywhere. But particles don't exist everywhere at once. On the contrary, they're what we call localized, or confined to a particular region of space.

The simplest way to visualize this is to impose some sort of boundary conditions: some region of space that can be different from purely empty space.

Trajectories of a particle in a box (also called an infinite square well) in classical mechanics (A) and quantum mechanics (B-F). In (A), the particle moves at constant velocity, bouncing back and forth. In (B-F), wavefunction solutions to the Time-Dependent Schrodinger Equation are shown for the same geometry and potential. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. (B,C,D) are stationary states (energy eigenstates), which come from solutions to the Time-Independent Schrodinger Equation. (E,F) are non-stationary states, solutions to the Time-Dependent Schrodinger equation.

Steve Byrnes / Sbyrnes321 of Wikimedia Commons

In our pre-quantum picture of the Universe, particles are simply points and nothing more: individual entities with a set of properties assigned to them. But we know that in the quantum Universe, we have to replace particles with wavefunctions, which are a probabilistic set of parameters that replace classical quantities like "position" or "momentum."

Instead of unique values, there are a set of possible values that a quantum field can take on. Some of the properties associated with a particle are continuous, like position, while others are discrete. The discrete ones are the most interesting in terms of fundamental particle properties, since those can only take on specific values that are defined by the characteristic conditions that the Universe sets out.

A guitar string, on its own, can vibrate in an infinite number of vibrational modes, corresponding to an unconstrained set of conceivable sounds. But by constraining the thickness of the string, the tension it's under, and the effective length of the part that vibrates, only a specific set of notes can emerge. These 'boundary conditions' are inseparable from the set of possible outputs.

Getty

A simple way to visualize this is to imagine a guitar. On a guitar, you have six strings of different thicknesses, where we can view thickness as a fundamental property of the string. If all you had were these strings (and no guitar), and you asked the question of the number of different possible ways these strings could vibrate, you'd wind up with an infinite number of allowable outcomes.

But guitars don't offer an infinite set of possibilities at all. We have boundary conditions on those strings:

the effective length of each string is constrained by the start-and-end points,

the number of possible excitations are constrained by the positions of the frets on the fretboard,

the vibrational modes are constrained by geometry and the music of overtones,

and the possible sounds it can make are constrained by the tension of each string.

These properties are uniquely determined by the size, string properties, and tuning of each individual guitar.

The Standard Model Lagrangian is a single equation encapsulating the particles and interactions of the Standard Model. It has five independent parts: the gluons (1), the weak bosons (2), how matter interacts with the weak force and the Higgs field (3), the ghost particles that subtract the Higgs-field redundancies (4), and the Fadeev-Popov ghosts, which affect the weak interaction redundancies (5). Neutrino masses are not included. Also, this is only what we know so far; it may not be the full Lagrangian describing 3 of the 4 fundamental forces.

Thomas Gutierrez, who insists there is one 'sign error' in this equation

In the case of our Standard Model particles, there are also a finite set of possibilities. They arise from a specific type of quantum field theory: a gauge theory. Gauge theories are invariant under a slew of transformations (like speed boosts, position translations, etc.) that our physical laws should also be invariant under.

The Standard Model in particular comes from a quantum field theory made up of three groups (as in the mathematics of Lie groups) all tied together:

SU(3), a group that's made of 3 × 3 matrices, which describes the strong interaction,

SU(2), a group that's made of 2 × 2 matrices, which describes the weak interaction,

and U(1), known as the circle group and made of all complex numbers with an absolute value of 1, which describes the electromagnetic interaction.

This diagram displays the structure of the standard model (in a way that displays the key relationships and patterns more completely, and less misleadingly, than in the more familiar image based on a 4x4 square of particles). In particular, this diagram depicts all of the particles in the standard model (including their letter names, masses, spins, handedness, charges, and interactions with the gauge bosons -- i.e. with the strong and electroweak forces). It also depicts the role of the Higgs boson, and the structure of electroweak symmetry breaking, indicating how the Higgs vacuum expectation value breaks electroweak symmetry, and how the properties of the remaining particles change as a consequence.

Latham Boyle and Mardus of Wikimedia Commons

The Standard Model isn't just a set of laws of physics, but provides proverbial boundary conditions that describe the spectrum of particles that can exist. Because the Standard Model isn't just made of a single quantum field in isolation, but all of the fundamental ones (except gravity) working together, the spectrum of particles we wind up with has a fixed set of properties.

This is determined by the specific mathematical structure — SU(3) × SU(2) × U(1) — that underlies the Standard Model. Each particle corresponds to the fundamental quantum fields of the Universe all excited in a particular way, with explicit couplings to the full suite of fields. This determines their particle properties, like:

mass,

electric charge,

color charge,

weak hypercharge,

lepton number,

baryon number,

lepton family number,

and spin.

The pattern of weak isospin, T_3, and weak hypercharge, Y_W, and color charge of all known elementary particles, rotated by the weak mixing angle to show electric charge, Q, roughly along the vertical. The neutral Higgs field (gray square) breaks the electroweak symmetry and interacts with other particles to give them mass.

Cjean42 of Wikimedia Commons

If the Standard Model were all there were, no other combinations would be allowed. The Standard Model gives you fermion fields, which correspond to the matter particles (quarks and leptons), as well as boson fields, which correspond to the force-carrying particles (gluons, weak bosons, and photon), as well as the Higgs.

The Standard Model was built with a set of symmetries in mind, and the particular ways these symmetries break determine the spectrum of allowed particles. They still require us to put in the fundamental constants that determine the specific values of particle properties, but the generic properties of a theory with:

6 quarks and antiquarks with three colors each,

3 charged leptons and antileptons,

3 neutrinos and antineutrinos,

8 massless gluons,

3 weak bosons,

1 massless photon,

and 1 Higgs boson,

are determined by the Standard Model itself.

The Standard Model of particle physics accounts for three of the four forces (excepting gravity), the full suite of discovered particles, and all of their interactions. Whether there are additional particles and/or interactions that are discoverable with colliders we can build on Earth is a debatable subject, but one we'll only know the answer to if we explore past the known energy frontier.

Contemporary Physics Education Project / DOE / NSF / LBNL

So how do we get quantum particles with the properties we do? Three things come together:

We have the laws of quantum field theory, which describe the fields permeating all of space that can be excited to different characteristic states.

We have the mathematical structure of the Standard Model, which dictates the allowable combinations of field configurations (i.e., particles) that can exist.

We have the fundamental constants, which provide the values of specific properties to each allowable combination: the properties of each particle.

And there may be more. The Standard Model may describe reality extremely well, but it doesn't include everything. It doesn't account for dark matter. Or dark energy. Or the origin of the matter-antimatter asymmetry. Or the reasons behind the values of our fundamental constants.

The Standard Model only provides the allowable configurations we know of. If neutrinos and dark matter are any indication, there ought to be more. One of the prime goals of 21st century science is to find out what else is there. Welcome to the cutting-edge frontier of modern physics.